Well, everybody who comes to Rome for the first time wants to see Piazza di Spagna (Spanish Steps). Those beautiful staircases dominated by the Trinità dei Monti church, the azaleas (only from mid-April to mid-May), the fountain, the people, the street vendors... it's dizzying. And although beautiful, it is not my favourite part of Rome. Even during low season the place is mobbed.
Although the Trinità dei Monti church and Piazza di Spagna has been paid for by the French, the square has been named after the Spanish embassy to the Pontifical States that was based here.
To me, the steps and the square are best approached from the church, either coming from Piazza Barberini or the Pincio Garden. The Trinità dei Monti is a beautiful baroque-style church built mainly in the 16th century and designed by Carlo Maderno. It's a great stop to take a break from the Roman sun and look at the painting (search for the "Assumption" by da Voltera, in which this pupil of Michelangelo painted his master in the scene). The square and steps were built later by architect Francesco de Sanctis between 1723 and 1725. In front of the church, you will notice many street vendors selling anything: painting, jewelry, food and drinks, postcards... Admire the view from the balustrade and take in the steps... you'll realize how crowded it is.
Now, avoid walking on people's hand or feet when going down. It used to be a place for artists and models to be picked up and well... it's still a place to be picked up and to see and be seen Just give in, buy a gelato, and enjoy!
At the end of the staircase is Fontana della Barcaccia, the Little Boat Fountain. It is not known which one of the Berninis (Pietro, the father, or Gianlorenzo, the more famous son) created the fountain (maybe they both worked on it). You'll often find lots of people there cooling off. The water is drinkable and it's a good idea to refill.
Resource for Tourists:
American Express has its office here.
Although I don't recommend eating there (come on, you're in Rome), check out the interior of the McDonald's. They established themselves in an old palazzo, causing the ire of the Romans.
Have a tea at Babington's or Caffe del Greco (on swanky via Condotti... check out the autograph by Buffalo Bill), where sophisticated British ladies and other rich foreigners were indulging during their mandatory pilgrimage to Rome.
From there, hit the designer shops of Via Condotti (if you have the wallet for it) or Via del Corso (more democratic), or reach the Trevi Foutain.